Chronoamperometric Sizing and Locating of Spherical Particles via

May 5, 2007 - Ian Streeter, Lei Xiao, Gregory G. Wildgoose, and Richard G. Compton ... Sinéad M. Matthews , Kamran Yunus , Adrian C. Fisher , Richard...
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J. Phys. Chem. C 2007, 111, 7801-7807

7801

Chronoamperometric Sizing and Locating of Spherical Particles via Potential-Step Transients Nicole Fietkau,† GuoQing Du,‡ Sine´ ad M. Matthews,‡ Michael L. Johns,‡ Adrian C. Fisher,‡ and Richard G. Compton*,† Physical and Theoretical Chemistry Laboratory, UniVersity of Oxford, South Parks Road, Oxford, OX1 3QZ, UK, and Department of Chemical Engineering, UniVersity of Cambridge, New Museums Site, Pembroke Street, Cambridge CB2 3RA, UK ReceiVed: January 18, 2007; In Final Form: April 2, 2007

The precise size of a spherical glass particle, its distance from the center of a microdisk electrode, and its exact location within an array of three noncollinear microdisk electrodes are determined by using simple Cottrellian-like potential-step chronoamperometric experiments. To first elucidate the size of a glass sphere and its distance from the center of the electrode, the sphere was positioned at arbitrary distances between the center of the sphere and the center of the electrode and a potential-step transient was recorded. The size of the sphere and the center-to-center distance is then deduced by comparison of the experimental current-time response with lattice Boltzmann simulations. The radius of the sphere and its distance from the center of the electrode is then confirmed, independently, by a microscopic measurement of the radius of the sphere and the center-to-center distance at the time of the electrochemical measurement. Second, the location of the sphere is elucidated in a so-called “triangulation” experiment in which the sphere was positioned in an array of three microdisk electrodes; the current-time response for each electrode was recorded individually and then compared with simulations. Excellent agreement between experiment and independent direct observation was found.

1. Introduction We have recently shown that cyclic voltammetric measurements as a function of voltage-scan rate can provide a very sensitive measurement of the radius of a sphere positioned in the center of a microelectrode.1 We also reported that the size and shape of microdroplets of dodecane of approximately 5 µm diameter immobilized on a regular array of hydrophobic polymer blocks of a partially blocked electrode can be determined by comparison of a simple Cottrellian-like potential-step experiment with simulations.2 In addition, the average size of inert particles has been measured by using a simple electrochemical procedure in which particles were deposited on a macroelectrode and cyclic voltammograms of a simple redox couple in solution were recorded as a function of voltage-scan rate. The accurate size of the inert particles was then determined by comparison of the experimental voltammogram with simulations for each individual voltage-scan rate.3 These approaches extend the use of electrochemical methods for the determination of shape and size, building on the ideas implicit in scanning electrochemical microscopy and those developed for the measurement of thin films.4,5 In the present article, we consider an approach for sizing a single particle of known shape (sphere) positioned at different distances, dc-c, between the center of the microdisk electrode of radius, re, and the center of the inert sphere of radius, rs, (Figure 1) by using a simple potential-step experiment conducted at the electrode. In particular, we use the nature of mass transport properties of microdisk electrodes to determine the size and shape of the particle by comparing the experimental results with * To whom correspondence should be addressed. E-mail: richard. [email protected]. Tel: +44 (0) 1865 275 413. Fax: +44 (0) 1865 275 410. † University of Oxford. ‡ University of Cambridge.

Figure 1. Schematic illustrating the experimental setup, in which re is the radius of the microdisk electrode, rs is the radius of the inert sphere and dc-c is the distance from the center of the electrode to the center of the sphere.

simulations. Additionally and novelly, we are able to deduce the exact location together with the radius of the sphere in a “triangulation” experiment, in which a glass sphere is positioned in an array of three microdisk electrodes (Figure 2a,b). The current-time transient for each individual electrode is recorded and then compared with simulations. 2. Theory 2.1. Lattice Boltzmann Theory. In this paper, we use a 3Dlattice Boltzmann method to study the one-electron electrolysis of species A at a microdisk electrode

A ( e- f B In each simulation, the potential is set to a value where no electrolysis current flows; the potential is then stepped to a value where oxidation/reduction occurs under transport-limited conditions. It is assumed that sufficient background electrolyte is present so that migration effects do not need to be considered. The lattice Boltzmann method involves tracking particle movement using a distribution function, fi(x, t), and these

10.1021/jp070429d CCC: $37.00 © 2007 American Chemical Society Published on Web 05/05/2007

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Figure 5. Theoretical study for a sphere of 40 µm radius positioned at different center-to-center distances from microdisk electrode of 40 µm radius. The parameters used for the simulations were as follows: ∆x ) 10-5 m, ∆t ) 0.002 s, NX ) 100, NY ) 80, NZ ) 100.

Figure 2. (a) Schematic diagram illustrating the experimental setup of the triangulation experiment, in which e1, e2, and e3 are microdisk electrodes of radius re, rs is the radius of the inert sphere, and de1-e2, de1-e3 and de2-e3 are the distances between the microdisk electrodes. (b) Top view of the experimental setup, in which dc-c1, dc-c2, and dc-c3 are the distances between the center of the electrode and the center of the sphere. Figure 6. Theoretical study for different radii of a sphere positioned at a distance of 60 µm from a microdisk electrode of 40 µm radius. The parameters used for the simulations were as follows: ∆x ) 10-5 m, ∆t ) 0.002 s, NX ) 100, NY ) 80, NZ ) 100. Figure 3. Schematic of the half way bounce-back boundary condition.

lattice Bhatnagar-Gross-Krook (LBGK) collision operator is used to give the lattice Boltzmann equation

fR,s (x + ∆x,t + 1) - fR,s (x,t) ) -

1 (eq) (f (x,t) - f R,s (x,t)) ks R,s (R ) 0,1,2,3,4,5,6,7) (1)

(eq) where ks is the collision relaxation time and f R,s (x,t) is the equilibrium distribution function. ∆x and ∆t are the lattice constant and time-step size, respectively. The effect of collision is to cause fR,s (x,t) to tend toward fR,s (x,t) so that the righthand side of above equation becomes zero leading to a steady state. The relaxation parameter is dependent on the diffusivity of the fluid and the distribution function is reliant on both the particle density, F, and particle momentum, Fu

∑R fR,s (x,t) ) F

(2)

∑R fR,s (x,t)eR ) Fu

(3)

For each species, s, the mass transfer equilibrium distribution function is given by eq 46 Figure 4. Schematic of the curved surface boundary condition.

distribution functions move from node to node on an ordered lattice in direction i (i ) 0, 1, ...,b), where b is the number of discrete particle velocities at each node. Starting from an initial state, the distribution functions evolve in two distinct stages of (1) streaming, in which fi(x, t) moves in direction i, and (2) collision, in which fi(x, t) is altered by some collision rule. The

(eq) f R,s (x,t) ) F(JR,s + KR,seRu)

(4)

with eR ) (0,0,0) R)0 rest channel ((1,0,0),(0, (1,0),(0,0, (1) R ) 1,2,3,4,5,6 non rest channel (5)

(

)

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Figure 7. Microscope image of the naked electrode of radius, re, and the electrode with a glass sphere of radius, rs, positioned at different centerto-center distances, dc-c. (a) Naked electrode with re ) 60 µm; (b) re ) 60 µm, rs ) 140 µm, and dc-c ) 180 µm; (c) re ) 60 µm, rs ) 130 µm, and dc-c ) 120 µm; and (d) re ) 60 µm, rs ) 150 µm, and dc-c ) 150 µm. Note that the electrode radius is slightly distorted due to the different diffraction indices of water and glass.

JR,s and Knonrestchannel,s are particularly chosen constants; Knonrestchannel,s is equal to one-half for this work and JR,s is determined by considering the following equation: (eq) (x,t) ∑R fR,s(x,t) ) ∑R fR,s

(6)

As the mass transfer simulations use lattices of only one speed, all of the nonrest channels share the same coefficient value. The lattice diffusivity is given by eq 7; the real diffusivity is related to the lattice diffusivity by eq 8

(

Dlattice ) Ca(1 - Jrest channel) ks Dlattice∆x2 Dreal ) ∆t

1 2

)

(7) (8)

where Ca is a lattice dependent coefficient equal to one-half and one-third for 2D and 3D simulations, respectively. Jrest channel must have a value between 0 and 1; in this work a value of one-fourth is used. 2.2. Definition of Boundary Conditions. In this work, two different boundary conditions have been applied; one where the physical boundary sits exactly on the mid-grid, such as a wall, and the other for the surface of the spherical obstruction where the physical boundary is a curved surface. In the first instance, the halfway bounce-back boundary condition is used, which is considered to have second-order accuracy.7 As depicted in Figure 3, three populations of the extra node receive another three populations respectively from the mirror conjugate sites in the fluid. This boundary condition was applied at the bottom wall and at the bounding walls of the simulation domain, which was expanded to ensure that the concentration profile did not extend to these walls. The implementation of the potential-step method means the con-

Figure 8. Influence of the sphere radius, rs, and the center-to-center distance, dc-c, on the current-time response corresponding to Figure 7. (a) I versus t plot and (b) enlargement of the experimental currenttime response at longer timescales for different rs and dc-c.

centration of reactant A on the electrode surface within the bottom wall is set to zero corresponding to transport-limited conditions. In the second case, the surface of sphere is regarded as a series of curves and the position of the surface of a sphere on the grid can be traced using eq 9

x2 + y2 ) rs2

(9)

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Figure 9. Experimental and simulated current-time response of the naked and the electrode with the glass spheres positioned at different centerto-center distances corresponding to Figure 7: (a) naked electrode with re ) 60 µm; (b) re ) 60 µm, rs ) 140 µm, and dc-c ) 180 µm; (c) re ) 60 µm, rs ) 130 µm, and dc-c ) 120 µm; and (d) re ) 60 µm, rs ) 150 µm, and dc-c ) 150 µm. The parameters used for the simulation are as follows: ∆x ) 10-5 m, ∆t (a) ) 0.0005 s, ∆t (b-d) ) 0.002 s, NX ) 100, NY ) 80, NZ ) 100.

where rs is the radius of the sphere. To solve this problem, we can rewrite eq 1 in terms of the collision and streaming steps. Collision step:

˜f R,s(x,t) - fR,s(x,t) ) -

1 (eq) (f (x,t) - fR,s (x,t)) ks R,s (R ) 0,1,2,3,4) (10)

Streaming step:

fR,s(x + ∆x,t + ∆t) ) ˜f R,s(x,t)

(11)

where fR,s(x,t) and ˜fR,s(x,t) denote the pre- and postcollision state of the distribution function, respectively. Illustrated in Figure 4, eq 11 requires ˜fR,s(x,t) at Xb on the solid node. The distribution function fR,s(x,t) at Xf moves to the solid node at Xb along the direction of eR to give the information of ˜fR,s(x,t) at Xb, then ˜fR,s(x,t) scatters back to the fluid node at Xf along the direction of eR . ˜fR,s(x,t) at Xb is related with the position of wall between the fluid and solid node; therefore a parameter, ∆, suggested by Mei8 is defined in eq 12

∆)

|Xf - Xw| |Xf - Xb|

(12)

To finish the streaming step

fjR,s (xf,t + ∆t) ) ˜f jR (xb,t)

(13)

By using the linear interpolation suggested by Filippova,9 the right-hand side can be written in the stagnant solution as

fjR (xb,t) ) (1 - χ)fR(xf,t) + χf/R(xb,t)

(14)

where f/R(xb,t) ) fR(eq)(xf,t), χ ) (2∆ - 1)k when ∆ g 1/2 or χ ) (2∆ - 1)(k - 1) when ∆ < 1/2. The electrolysis current, ie, can be calculated using eq 15

ie ) -nFDAJe

(15)

where n is the number of electrons transferred per reaction, F is the Faraday constant, D is the diffusion coefficient, A is the electrode area and Je is the concentration flux at the surface of the electrode. 3. Experimental Section 3.1. Chemical Reactants and Instrumentation. All chemicals employed were of analytical grade and used as received without any further purification. Potassium hexacyanoferrate(II)trihydrate (K4Fe(CN)6) was purchased from Lancaster Synthesis and potassium chloride (KCl) was supplied by Riedelde-Hae¨n. All solutions were prepared with deionized water with resistivity of no less than 18.2 MΩ cm (Millipore water systems, U.K.). The acid-washed glass beads were between 212 and 300 µm in diameter and obtained from Sigma-Aldrich. Electrochemical measurements were carried out by using a µ-Autolab II (ECO-Chemie, Utrecht, Netherlands) potentiostat interfaced to a PC by using GPES (version 4.9) software for Windows. All measurements were conducted by using a threeelectrode cell, in which the working electrode was a homemade platinum microdisk electrode. The counter electrode was a bright platinum wire, and a saturated calomel electrode (SCE) was employed as a reference electrode. All experiments were carried out at 298 ( 2 K. Before measurements were taken, the working electrode was polished with decreasing sizes of alumina slurry (1-0.3 µm) on soft lapping polishing pads and then thoroughly rinsed with pure water.

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Figure 10. Comparison between experimental and theoretical results of the percent decrease of I at t ) 10 s of the potential-step experiment. (a) re ) 60 µm, rs ) 140 µm, and dc-c ) 60 µm; (b) re ) 60 µm, rs ) 140 µm, and dc-c ) 90 µm; (c) re ) 60 µm, rs ) 130 µm, and dc-c ) 120 µm; (d) re ) 60 µm, rs ) 150 µm, and dc-c ) 150 µm; and (e) re ) 60 µm, rs ) 140 µm, and dc-c ) 180 µm. The parameters used for the simulations were as follows: ∆x ) 10-5 m, ∆t ) 0.002 s, NX ) 100, NY ) 80, NZ ) 100. Figure 12. Influence of the sphere radius, rs, and the center-to-center distance, dc-c, on the current-time response corresponding to Figure 11. (a) I versus t plot and (b) enlargement of the experimental currenttime response at longer timescales for different rs and dc-c.

Figure 11. (a) Microscope image of the naked electrodes and (b) microscope image of the electrode with a glass sphere positioned between the electrodes where re ) 20 µm, rs ) 110 µm, dc-c1 ) 150 µm, dc-c2 ) 80 µm, and dc-c3 ) 110 µm. Note that the electrode radii and positions are slightly distorted due to the different diffraction indices of water and glass.

Optical microscope images were recorded by using a Digital Instruments OMV-PAR microscope based on a Sony XC-999P CCD camera, which has a maximum resolution of 752 × 582 pixels over an area of 540 × 400 µm. 3.2. Fabrication of the Microdisk Electrodes and Positioning of the Glass Sphere. The microelectrode consisted of a

platinum wire sealed into an epoxy polymer. The fabrication procedure was as follows: Expoxy resin (Epotek H77A, Promatech Ltd, Cirenchester, U.K.) and hardener (Epotek H77B, Promatech Ltd, Cirenchester, U.K.) were mixed manually in the ratio of resin:hardener 20:3 with a spatuala. A plate of epoxy of the dimensions of 1.0 cm × 1.0 cm × 0.5 cm was fabricated and the 1.0 cm × 1.0 cm surface was polished to a flat surface with abrasive paper. Half of the polished surface was covered with a thin layer of epoxy, in which a 50 µm platinum microwire (99.99%, Goodfellows, Cambridge, U.K.) was placed and dried at 80 °C for several hours. Before the metal wire was covered with a second layer of epoxy polymer of 0.5 cm thickness, the electrical connection was made. The electrode was cured at 80 °C in the oven; then, its working face was ground down and polished by first using abrasive paper then decreasing sizes of alumina slurry (25-0.3 µm).10 The fabrication of the electrode array of three platinum microdisk electrodes of 25 µm radius was similar to the procedure described above; initially two platinum wires were placed very close to each other on the plate of epoxy under a microscope before covering them with a thin layer of epoxy and positioning the third wire in the middle of the first two wires. The electrode radius was calibrated electrochemically by using a 3 mM K4Fe(CN)6/0.1 M KCl aqueous solution and by simulating the resulting current responses for various scan rates with a microdisk simulation program with a diffusion coefficient of 0.63 × 10-5 cm2 s-1 and a standard heterogeneous rate constant of 0.05 cm s-1 at a temperature of 298 K, which are in good agreement with the reported values.11,12 The glass spheres were immersed into 3 mM K4Fe(CN)6/0.1 M KCl aqueous solution, deposited with a glass pipet onto the electrode surface, and placed in the center of the electrode by using the tip of a very thin glass pipet mounted on a x-y-z micropositioning device. 4. Results and Discussion The influence of the radius of a sphere, rs, positioned at different center-to-center distances, dc-c, on the electrochemical response of a microdisk electrode was studied theoretically and

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Figure 13. Experimental and simulated current-time response of the electrode with the glass spheres positioned at different center-to-center distances corresponding to Figure 11. (a) re ) 20 µm, rs ) 110 µm, and dc-c1 ) 150 µm; (b) re ) 20 µm, rs ) 110 µm, and dc-c2 ) 80 µm; and (c) re ) 20 µm, rs ) 110 µm, and dc-c3 ) 110 µm. The parameters used for the simulations were as follows: ∆x ) 10-5 m, ∆t ) 0.002 s, NX ) 100, NY ) 80, NZ ) 100.

Figure 14. Comparison between experimental and theoretical results of the percent decrease of I at t ) 10 s of the potential-step experiment. (a) re ) 20 µm, rs ) 110 µm, and dc-c ) 80 µm; (b) re ) 20 µm, rs ) 110 µm, and dc-c ) 110 µm; (c) re ) 20 µm, rs ) 120 µm, and dc-c ) 117 µm; (d) re ) 20 µm, rs ) 130 µm, and dc-c ) 132 µm; and (e) re ) 20 µm, rs ) 110 µm, and dc-c ) 150 µm. The parameters used for the simulation are as follows: ∆x ) 10-5 m, ∆t ) 0.002 s, NX ) 100, NY ) 80, NZ ) 100.

experimentally (see Figures 1 and 2). The electrochemical system is perturbed by applying an instantaneous potential step from a region of no Faradaic processes to a point at which the one-electron oxidation of the electroactive species A occurs at a diffusion-controlled rate and the resulting current response is monitored as a function of time. Initially, only species A is present in the solution; after the potential step, the concentration of A at the electrode surface decreases to zero and a concentration gradient between the interfacial region and the bulk solution is established. As molecules of A diffuse to the electrode surface and are converted to the reduced species B, the diffusion layer extends further into the solution. The presence of the sphere

adjacent to the microdisk electrode modifies the mass-transport properties of the system as has been shown in previous publications.1,13-15 Figure 5 depicts the percentage decrease of the current at t ) 10 s for a sphere of 40 µm radius located at different distances to a microdisk of 40 µm radius. As the center-tocenter distance decreases below 40 µm, parts of the sphere start to sit directly above of the electrode; accordingly, the growth of the diffusion layer is blocked to a certain extent. However, when dc-c is greater than 40 µm, the diffusion layer can extend further into solution and becomes progressively like the diffusion layer when there is no sphere present. This results in the sensitivity of the current response changing when the sphere sits directly above the electrode surface. A similar theoretical study was carried out to elucidate the influence of the sphere radius at a given center-to-center distance on the electrochemical response (Figure 6). As expected, the current decreases quicker with increasing sphere radius. In summary, Figures 5 and 6 show that the current-time response is responsive to both the sphere position and size. Figure 7a shows a microscope image of the isolated microdisk electrode, and Figure 7b-d shows images of the electrode modified with glass spheres of different radii positioned at different center-to-center distances. Analysis of the microscopic images resulted in an electrode radius, re, of 60 µm and sphere radii, rs, of 140 ( 2 µm, 120 ( 2 µm, and 150 ( 2 µm, respectively. For all cases, current-time responses resulting from a potential step from -0.1 to 0.4 V for the oxidation of 3 mM ferrocyanide/0.1 M KCl in aqueous solution were recorded. Figure 8a,b illustrate the influence of the sphere radius and the center-to-center distance on the current-time response. A

Sizing and Locating of Spherical Particles decrease in the center-to-center distance results in a quicker decay of the current compared to the naked electrode as there is less material near the electrode to electrolyze (Figure 8b). Figure 9 shows the comparison of the experimental and simulated current-time responses for both the electrode and the electrode with the spheres positioned at different center-tocenter distances corresponding to Figure 7. Figure 10 shows the percentage decrease of the current at t ) 10 s for different sphere sizes and center-to-center distances. The excellent agreement between theory and experiment confirms the values of rs and dc-c measured with the help of the microscope, hence giving physical support for the applicability of sizing and, implicitly, shaping particles in an electrode array. Figure 11a,b illustrate a triangulation experiment, where a sphere of a radius of 110 µm is positioned in an array of three microdisk electrodes. Analysis of the microscopic images resulted in re ) 20 µm, rs ) 110 ( 2 µm, drse1 ) 150 ( 2 µm, drse2 ) 80 ( 2 µm, and drse3 ) 110 ( 2 µm. The current-time response, recorded separately for each individual electrode, clearly shows the influence of the center-to-center distance on the current-time response (Figure 12a,b). A decrease in dc-c results in a quicker decay of the current because there is less material near the electrode to be electrolyzed. The good fit between experiment and theory of the current-time response and the percent decrease of the current at t ) 10 s in Figures 13 and 14, respectively, confirm that the triangulation experiment not only allows us to determine the size and confirm the shape of the sphere and the distance from the center of the electrode to the center of the sphere (relative position of the sphere) but also the absolute location of the sphere within the electrode array. This experiment therefore provides proof of concept and support of the voltammetric “locating” of static particles. To extend this concept of voltammetrically sizing and locating spherical particles to those of unknown size or location, the system would require a calibration of the current response for spheres of known size and location. 5. Conclusion By positioning glass spheres at various distances to a microdisk electrode, we have demonstrated that the electro-

J. Phys. Chem. C, Vol. 111, No. 21, 2007 7807 chemical current-time response can be used to determine the size of the spherical particle. We have also shown that the absolute position of a spherical particle can be elucidated by using the electrochemical response of three microdisk electrodes. Furthermore, we are confident that this approach can be used to size spherical particles in the nanometer range if the electrodes radius and the center-to-center distance is adjusted accordingly. Work to extend this concept to measure particles of different shapes is currently on the way in our labs, as well as the tracking of moving particles. Acknowledgment. N.F. thanks the EPSRC for a studentship. References and Notes (1) Fietkau, N.; Chevallier, F. G.; Li, J.; Jones, T. G. J.; Compton, R. G. ChemPhysChem 2006, 7, 2162. (2) Barnes, A. S.; Fietkau, N.; Chevallier, F. G.; del Campo, J.; Mas, R.; Mun˜oz, F. X.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2006, 602, 1. (3) Davies, T. J.; Lowe, E. R.; Wilkins, S. J.; Compton, R. G. ChemPhysChem 2005, 6, 1340. (4) Kwak, J.; Bard, A. J. Anal. Chem. 1989, 61, 1794. (5) Arkoub, I. A.; Amatore, C.; Sella, C.; Thouin, L.; Warkocz, J.-S. J. Phys. Chem. B 2001, 105, 8694. (6) Flekkøy, E. G. Phys. ReV. E: Stat., Nonlinear, Soft Matter Phys. 1993, E47, 4247. (7) Succi, S. The Lattice Boltzmann Equation for Fluid Dynamics and Beyond; Oxford University Press: New York, 2001. (8) Mei, R.; Luo, L.-S.; Shyy, W. J. Comput. Phys. 1999, 155, 307. (9) Filippova, O.; Ha¨nel, D. J. Comput. Phys. 1998, 147, 219. (10) Maisonhaute, E.; Del Campo, F. J.; Compton, R. G. Ultrason. Sonochem. 2002, 9, 275. (11) Stackelberg, M. V.; Pilgram, M.; Toome, V. Z. Elektrochem. Angew. Phys. Chem. 1953, 57, 342. (12) Bard, A. J.; Faulkner, L. R. Electrochemical Methods, second edition; John Wiley and Sons: New York, 2001. (13) Chevallier, F. G.; Davies, T. J.; Klymenko, O. V.; Jiang, L.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2005, 580, 265. (14) Chevallier, F. G.; Davies, T. J.; Klymenko, O. V.; Jiang, L.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2005, 577, 211. (15) Chevallier, F. G.; Fietkau, N.; del Campo, J.; Mas, R.; Munoz, F. X.; Jiang, L.; Jones, T. G. J.; Compton, R. G. J. Electroanal. Chem. 2006, 596, 25.